Quantum Stochastic Maps and Frobenius Perron Theorem

Quantum Stochastic Maps and Frobenius Perron Theorem Karol Życzkowski in collaboration with W. Bruzda, V. Cappellini, H.-J. Sommers, M. Smaczyński Institute of Physics, Jagiellonian University, Cracow, Poland and Center for Theoretical Physics, Polish Academy of Sciences KCIK Sopot, November 30, 2012 KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 1 / 27

Pure states in a finite dimensional Hilbert space H N Qubit = quantum bit; N = 2 ψ = cos ϑ 2 1 + eiφ sin ϑ 2 0 P 1 = Bloch sphere for N = 2 pure states Space of pure states for an arbitrary N: a complex projective space P N 1 of 2N 2 real dimensions. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 2 / 27

Unitary evolution Fubini-Study distance in P N 1 D FS ( ψ, ϕ ) := arccos ψ ϕ Unitary evolution Let U = exp(iht). Then ψ = U ψ. Since ψ ϕ 2 = ψ U U ϕ 2 any unitary evolution is a rotation in P N 1 hence it is an isometry (with respect to any standard distance!) Classical limit: what happend for large N? How an isometry may lead to classically chaotic dynamics? The limits t and N do not commute. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 3 / 27

Interacting Systems & Mixed Quantum States Set M N of all mixed states of size N M N := {ρ : H N H N ; ρ = ρ, ρ 0, Trρ = 1} example: M 2 = B 3 Ê 3 - Bloch ball with all pure states at the boundary The set M N is compact and convex: ρ = i a i ψ i ψ i where a i 0 and i a i = 1. It has N 2 1 real dimensions, M N Ê N2 1. How the set of all N = 3 mixed states looks like? An 8 dimensional convex set with only 4 dimensional subset of pure (extremal) states, which belong to its 7 dim boundary KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 4 / 27

Quantum maps Quantum operation: linear, completely positive trace preserving map positivity: Φ(ρ) 0, ρ M N complete positivity: [Φ ½ K ](σ) 0, σ M KN and K = 2, 3,... Enviromental form (interacting quantum system!) ρ = Φ(ρ) = Tr E [U (ρ ω E ) U ]. where ω E is an initial state of the environment while UU = ½. Kraus form ρ = Φ(ρ) = i A i ρa i, where the Kraus operators satisfy i A i A i = ½, which implies that the trace is preserved. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 5 / 27

Classical probabilistic dynamics & Markov chains Stochastic matrices Classical states: N-point probability distribution, p = {p 1,...p N }, where p i 0 and N i=1 p i = 1 Discrete dynamics: p i = S ij p j, where S is a stochastic matrix of size N and maps the simplex of classical states into itself, S : N 1 N 1. Frobenius Perron theorem Let S be a stochastic matrix: a) S ij 0 for i, j = 1,...,N, b) N i=1 S ij = 1 for all j = 1,...,N. Then i) the spectrum {z i } N i=1 of S belongs to the unit disk, ii) the leading eigenvalue equals unity, z 1 = 1, iii) the corresponding eigenstate p inv is invariant, Sp inv = p inv. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 6 / 27

Quantum stochastic maps (trace preserving, CP) Superoperator Φ : M N M N A quantum operation can be described by a matrix Φ of size N 2, ρ = Φρ or ρ mµ = Φmµ nν ρ nν. The superoperator Φ can be expressed in terms of the Kraus operators A i, Φ = i A i Ā i. Dynamical Matrix D: Sudarshan et al. (1961) obtained by reshuffling of a 4 index matrix Φ is Hermitian, Dmn µν := Φmµ nν, so that D Φ = D Φ =: ΦR. Theorem of Choi (1975). A map Φ is completely positive (CP) if and only if the dynamical matrix D Φ is positive, D 0. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 7 / 27

Spectral properties of a superoperator Φ Quantum analogue of the Frobenious-Perron theorem Let Φ represent a stochastic quantum map, i.e. a ) Φ R 0; (Choi theorem) b ) Tr A Φ R = ½ k Φ kk = δ ij. (trace preserving condition) ij Then i ) the spectrum {z i } N2 i=1 of Φ belongs to the unit disk, ii ) the leading eigenvalue equals unity, z 1 = 1, iii ) the corresponding eigenstate (with N 2 components) forms a matrix ω of size N, which is positive, ω 0, normalized, Trω = 1, and is invariant under the action of the map, Φ(ω) = ω. Classical case In the case of a diagonal dynamical matrix, D ij = d i δ ij reshaping its diagonal {d i } of length N 2 one obtains a matrix of size N, where S ij = D ii, of size N which is stochastic and recovers the standard F P theorem. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 8 / 27 jj

Decoherence for quantum states and quantum maps Quantum states classical states = diagonal matrices Decoherence of a state: ρ ρ = diag(ρ) Quantum maps classical maps = stochastic matrices Decoherence of a map: The Choi matrix becomes diagonal, D D = diag(d) so that the map Φ = D R D R S where for any Kraus decomposition defining Φ(ρ) = i A iρa i the corresponding classical map S is given by the Hadamard product, S = i A i Āi If a quantum map Φ is trace preserving, i A i A i = ½ then the classical map S is stochastic, j S ij = 1. If additionally a quantum map Φ is unital, i A ia i = ½ then the classical map S is bistochastic, j S ij = i S ij = 1. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 9 / 27

Exemplary spectra of (typical) superoperators 1 Im(z) a) N = 2 1 Im(z) b) N = 3 0 0 Re(z) Re(z) 1 1 1 0 1 1 0 1 Spectra of several random superoperators Φ for a) N = 2 and b) N = 3 contain: i) the leading eigenvalue z 1 = 1 corresponding to the invariant state ω, ii) real eigenvalues, iii) complex eigenvalues inside the disk of radius r = z 2 1. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 10 / 27

Random (classical) stochastic matrices Ginibre ensemble of complex matrices Square matrix of size N, all elements of which are independent random complex Gaussian variables. An algorithm to generate S at random: 1) take a matrix X form the complex Ginibre ensemble 2) define the matrix S, S ij := X ij 2 / N X ij 2, which is stochastic by construction: each of its columns forms an independent random vector distributed uniformly in the probability simplex N 1. i=1 KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 11 / 27

Random quantum states How to generate a mixed quantum state at random? 1) Fix M 1 and take a N M random complex Ginibre matrix X; 2) Write down the positive matrix Y := XX, 3) Renormalize it to get a random state ρ, ρ := Y TrY. This matrix is positive, ρ 0 and normalised, Trρ = 1, so it represents a quantum state! Special case of M = N (square Ginibre matrices) Then random states are distributed uniformly with respect to the Hilbert-Schmidt (flat) measure, e.g. for M = N = 2 random mixed states cover uniformly the interior of the Bloch ball. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 12 / 27

Random (quantum) stochastic maps An algorithm to generate Φ at random: 1) Fix M 1 and take a N 2 M random complex Ginibre matrix X; 2) Find the positive matrix Y := Tr A XX and its square root Y ; 3) Write the dynamical matrix (Choi matrix) D = (½ N 1 Y )XX (½ N 1 Y ) ; 4) Reshuffle the Choi matrix to obtain the superoperator Φ = D R and use to produce a random map, ρ mµ = Φmµ ρ nν. nν Then Tr A D = ½, so the map Φ obtained in this way is stochastic! i.e. Φ is completely positive and trace preserving KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 13 / 27

Random stochastic maps II Probability distribution for random maps P(D) det(d M N2 ) δ(tr A D ½), In the special case M = N 2 the factor with the determinant vanishes, so there are no other constraints on the distribution of the random Choi matrix D, besides the partial trace relation, Tr A D = ½. Interaction with M dimensional enviroment 1 ) Chose a random unitary matrix U according to the Haar measure on U(NM) 2 ) Construct a random map defining ρ = Tr M [U(ρ ν ν )U ], where ν H M is an arbitrary (fixed) pure state of the environment. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 14 / 27

Bloch vector representation of any state ρ ρ = N 2 1 i=0 τ i λ i where λ i are generators of SU(N) such that tr ( λ i λ j) = δ ij and λ 0 = ½/ N and τ i are expansion coefficients. Since ρ = ρ, the generalized Bloch vector τ = [τ 0,...,τ N 2 1] is real. Stochastic map Φ in the Bloch representation The action of the map Φ can be represented as τ = Φ(τ) = Cτ + κ, where C is a real, asymmetric contraction [ ] matrix of size N 2 1 while κ is 1 0 a translation vector. Thus Φ = and the eigenvalues of C are κ C also eigenvalues of Φ. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 15 / 27

Random maps & random matrices Full rank, symmetric case M = N 2 For large N the measure for C can be described by the real Ginibre ensemble of non-hermitian Gaussian matrices. Spectral density in the unit disk The spectrum of Φ consists of: i) the leading eigenvalue z 1 = 1, ii) the component at the real axis, the distribution of which is asymptotically given by the step function P(x) = 1 2Θ(x 1)Θ(1 x), iii) complex eigenvalues, which cover the disk of radius r = z 2 1 uniformly according to the Girko distribution. Subleading eigenvalue r = z 2 The radius r is determined by the trace condition: Since the avearge TrD 2 = TrΦΦ const then r 1/N so the spectrum of the rescaled matrix Φ := NΦ covers the entire unit disk. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 16 / 27

Spectral density for random stochastic maps Numerical results a) Distribution of complex eigenvalues of 10 4 rescaled random operators Φ = NΦ already for N = 10 can be approximated by the circle law of Girko. b) Distribution of real eigenvalues P(x) of Φ plotted for N = 2, 3, 7 and 14 tends to the step function! KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 17 / 27

Convergence to equilibrium and decoherence rate Average trace distance to invariant state ω = Φ(ω) L(t) = Tr Φ t (ρ 0 ) ω ψ, where the average is performed over an ensemble of initially pure random states, ρ 0 = ψ ψ. Numerical result confirm an exponential convergence, L(t) exp( αt). a) Average trace distance of random pure states to the invariant state of Φ as a function of time for M = N 2 and N = 4( ), 6( ), 8( ). b) mean convergence rate α Φ scales as ln N = 1 2 lnm KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 18 / 27

Unistochastic Maps defined by an interaction of the ancilla of the same dimension initially in the maximally mixed state, ρ = Φ U (ρ) = Tr env [ U(ρ 1 N ½ N)U ] Unistochastic maps are unital, Φ U (½) = ½, hence bistochastic. Is every bistochastic map unistochastic? No! One qubit bistochastic maps = Pauli channels, ρ = Φ p (ρ) Φ p (ρ) = 3 i=0 p i σ i ρσ i, where p i = 1 while σ 0 = ½ and remaining three σ i denote Pauli matrices. Let η = (η 1, η 2, η 3 ) be the damping vector containing three axis of the ellipsoid - the image of the Bloch ball by a bistochastic map. Set B 2 of one-qubit bistochastic maps = regular tetrahedron with corners at η = (1, 1, 1), (1, 1, 1), ( 1, 1, 1), ( 1, 1, 1) corresponding to σ i with i = 0,...,3. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 19 / 27

One Qubit Unistochastic Maps Unistochastic maps do satisfy following restrictions for the damping vector η = (η 1, η 2, η 3 ) η 1 η 2 η 3, η 2 η 3 η 1, η 3 η 1 η 2. The set U 2 forms a (non-convex!) subset of the tetrahedron B 2 of bistochastic maps, Musz, Kuś Życzkowski, 2012 Example: The following Pauli channel Φ p (ρ) = 3 i=1 1 3 σ iρσ i, of rank three is not unistochastic. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 20 / 27

Interacting quantum dynamical systems Generalized quantum baker map with measurements a) Quantisation [ of Balazs and ] Voros applied for the asymmetric map B = F FN/K 0 N, where N/K Æ. 0 F N(K 1)/K where F N denotes the Fourier matrix of size N. Then ρ = Bρ i B b) M measurement operators projecting into orthogonal subspaces ρ i+1 = M i=1 P iρ P i c) vertical shift by /2, (Alicki, Loziński, Pakoński, Życzkowski 2004) Standard classical model K = 2, dynamical entropy H = ln2; Asymmetric model, K > 2, entropy decreases to zero as K. Classical limit: N with K N. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 21 / 27

Exemplary spectra of superoperator for L fold generalized baker map B L & measurement with M Kraus operators for N = 64 and M = 2: 1 a) 1 b) 1 c) 0.5 0.5 0.5 0 0 0 0.5 0.5 0.5 1 1 0.5 0 0.5 1 K = 64, L = 1 1 1 0.5 0 0.5 1 K = 4, L = 16 1 1 0.5 0 0.5 1 K = L = 32 a) weak chaos (K = 64 and L = 1), b) strong chaos (K = 4 and L = 4) universal behaviour: λ 1 = 1 and uniform disk of eigenvalues of radius R, c) weak chaos (K = 32 and L = 32). KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 22 / 27

Random quantum operation (stochastic maps) a) Interaction with M dim. enviroment (Cappellini et al. 2009) 1) Fix the size of ancilla M 1, 2) Chose a random unitary matrix U according to the Haar measure 3) Take a fixed pure state ν H M and construct a random map ρ = Tr M [U(ρ ν ν )U ], b) Random external fields (bistochastic) Φ R (ρ) = M p m U m ρu m m=1 with p m = 1, where U m are random Haar unitary matrices, U m U m = ½, m = 1,,,M. c) Projected Unitary Matrices Φ P (ρ) = M m=1 P muρu P m, where U is a random Haar unitary and P m = P m = P 2 m are projection operators on mutually orthogonal subspaces of the same dimension d = N/M. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 23 / 27 m

Exemplary spectra of superoperators for random operations acting N = 64 dim Hilbert space H 64 with M = 2 Kraus operators 1 a) 1 1 b) c) 0.5 im(λ) 0 0.5 0 0.5 0 0.5 0.5 0.5 1 1 0.5 0 0.5 1 re(λ) 1 1 0.5 0 0.5 1 re(λ) 1 1 0.5 0 0.5 1 re(λ) a) ensemble Φ E of random coupling with an environment, b) ensemble Φ R of Random external fields, c) ensemble Φ P of Projected Unitary Matrices this ensemble describes well the deterministic generalized quantum baker map (under the assumption of strong chaos and strong decoherence!) KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 24 / 27

Radius R of the essential spectrum If measurement is done by projection on M subspaces of the same size d = N/M the radius scales as R 1/ M. 1 1 Im λ 0 Im λ 0 1 1 0 1 Re λ 1 1 0 1 Re λ Spectra of superoperator of generalized baker map for N = 64, a) M = 2, R 1/ 2, b) M = 8, R 1/ 8, c) Radius R as a function of the number M of Kraus operators for baker map (+) and random operations ( ). The radius R determines the decoherence rate hskip 1.5cm and the speed of convergence to equilibrium! KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 25 / 27

Concluding Remarks I Quantum Chaos: a) in case of closed systems one studies unitary evolution operators and characterizes their spectral properties, b) for open, interacting systems one analyzes non unitary time evolution described by quantum stochastic maps. a link between a) open quantum systems b) ensembles of random operations c) real Ginibre ensemble of random matrices was established. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 26 / 27

Concluding Remarks II We analyzed spectral properties of quantum stochastic maps and formulated a quantum analogue of the Frobenius-Perron theorem. Universality of the spectra: in the case of i) strong chaos and ii) strong decoherence (e.g. projection onto M subspaces of the same size) the spectra consist of a leading eigenvalue λ 1 = 1 and other eigenvalues covering the disk of radius R 1/ M. Sequential action of a fixed map Φ brings exponentially fast all pure states to the invariant state ω = Φ(ω). The rate of convergence α is determined by modulus of the subleading eigenvalue R = λ 2 as α = lnr. In the case of weak decoherence the spectrum is non-universal. KŻ (IF UJ/CFT PAN ) Quantum Frobenius Perron November 30, 2009 27 / 27